Parallel Transport along Seifert Manifolds and Fractional Monodromy

N. Martynchuk and K. Efstathiou
Preprint, 2016

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We show that the Euler number of a Seifert manifold can be defined in terms of parallel transport of homology cycles along this manifold. Such point of view is motivated by the study of integrable Hamiltonian systems, where parallel transport of homology cycles is used to define various notions of monodromy a la Duistermaat. We use the relation between the Euler number and parallel transport to compute fractional monodromy in integrable 2 degrees of freedom systems with $\mathbb S^1$ symmetry.